Lectures on Symplectic Geometry by Ana Cannas da Silva
Book Contents :-
INTRODUCTION
PART I — SYMPLECTIC MANIFOLDS
1. Symplectic Forms
2. Symplectic Form on the Cotangent Bundle
PART II — SYMPLECTOMORPHISMS
3. Lagrangian Submanifolds
4. Generating Functions
5. Recurrence
PART III — LOCAL FORMS
6. Preparation for the Local Theory
7. Moser Theorems
8. Darboux–Moser–Weinstein Theory
9. Weinstein Tubular Neighborhood Theorem
PART IV — CONTACT MANIFOLDS
10. Contact Forms
11. Contact Dynamics
PART V — COMPATIBLE ALMOST COMPLEX STRUCTURES
12. Almost Complex Structures
13. Compatible Triples
14. Dolbeault Theory
PART VI — KÄHLER MANIFOLDS
15. Complex Manifolds
16. Kähler Forms
17. Compact Kähler Manifolds
PART VII — HAMILTONIAN MECHANICS
18. Hamiltonian Vector Fields
19. Variational Principles
20. Legendre Transform
PART VIII — MOMENT MAPS
21. Actions
22. Hamiltonian Actions
PART IX — SYMPLECTIC REDUCTION
23. The Marsden–Weinstein–Meyer Theorem
24. Reduction
PART X — MOMENT MAPS REVISITED
25. Moment Map in Gauge Theory
26. Existence and Uniqueness of Moment Maps
27. Convexity
PART XI — SYMPLECTIC TORIC MANIFOLDS
28. Classification of Symplectic Toric Manifolds
29. Delzant Construction
30. Duistermaat–Heckman Theorems
About this book :-
"Lectures on Symplectic Geometry" by "Ana Cannas da Silva" is a clear and well-structured introduction to "symplectic geometry", written for graduate students and mathematically mature readers. The book grows out of lecture notes and is designed to bridge the gap between basic "differential geometry" and modern research topics. It assumes familiarity with manifolds and calculus on forms, making it ideal for readers moving beyond introductory geometry.
The text develops the subject step by step, starting with symplectic forms and symplectic manifolds, and moving toward deeper ideas such as "Hamiltonian mechanics", moment maps, and symplectic reduction. Key results like Darboux’s theorem are presented with strong geometric intuition, helping readers understand not just the statements, but why they matter. The book also explores links to "Lie groups", contact geometry, and Kähler manifolds, showing how symplectic geometry fits into a broader mathematical framework.
What makes this book especially valuable is its balance between rigor and accessibility. The writing is concise and human-friendly, with examples and exercises that reinforce understanding. Overall, it is an excellent foundation for anyone planning further study or research in "modern geometry" or mathematically oriented physics.
Book Detail :-
Title:
Lectures on Symplectic Geometry by Ana Cannas da Silva
Publisher:
Springer
Year:
2006
Pages:
225
Type:
PDF
Language:
English
ISBN-10 #:
3540421955
ISBN-13 #:
978-3540421955
License:
External Educational Resource
Amazon:
Amazon
About Author :-
The author
Ana Cannas da Silva
is a "Portuguese mathematician" best known for her contributions to "symplectic geometry" and geometric topology. She earned her PhD from "MIT" under Victor Guillemin and has held academic positions at leading institutions, including Princeton and UC Berkeley. She is currently a professor at "ETH Zurich", where she teaches and researches modern geometry. Her book "Lectures on Symplectic Geometry" is widely used by graduate students and researchers, valued for its clear exposition and strong connection between "differential geometry", topology, and classical mechanics.
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